Computer Algebra

Computer Algebra Symbols as well as numbers can be manipulated by a computer.

New, general-purpose algorithms can undertake a wide variety

of routine mathematical work and solve intractable problems

by Richard Pavelle, Michael Rothstein and John Fitch

Of all the tasks to which the computer can be applied none is more daunting than the manipu

lation of complex mathematical expressions. For numerical calculations the digital computer is now thoroughly established as a device that can greatly ease the human burden of work. It is less generally appreciated that there are computer programs equally well adapted to the manipulation of algebraic expressions. In other words, the computer can work not only with numbers themselves but also with more abstract symbols that represent numerical quantities.

Perhaps one reason the algebraic capabilities of the computer have not been fully exploited is that computer programming itself is much like algebra in character. It might therefore seem there is a natural division of labor: the programmer manipulates algebraic symbols, whereas the computer is confined to arithmetic calculations, or "number crunching." Yet the dichotomy between the programmer and the computer was recognized to be a false one as early as 1844 by Augusta Ada Byron, Countess of Lovelace, the daughter of Lord Byron and possibly the first computer programmer. Lady Lovelace was a benefactor of Charles Babbage's and devised some of the programs for the early mechanical computer that Babbage called the difference engine. She pointed out that the computer could "arrange and combine its numerical quantities exactly as if they were letters or any other general symbols; and in fact it might bring out its results in algebraic notation, were provisions made accordingly." Similarly, the modern digital computer is a general-purpose machine that can carry out any algorithm, or precisely specified procedure. The algorithms of algebra can be executed by a computer as readily as those of arithmetic.

In 1973 one of us (Pave lie) undertook an algebraic calculation pertaining to the general theory of relativity; the calculation required three months of work with pencil and paper. The following year a more formidable problem arose in an attempt to define the mathematical

136

properties that might d istinguish general relativity from various other theories of gravitation. Instead of attempting another calculation by hand, Pavelle decided to construct a computer program for manipulating mathematical expressions of the kind that commonly appear in gravitational theories. The program was then written in the computer language of a powerful system of algebraic programs called MACSYMA, then under development at the Massachusetts Institute of Technology. Once Pavelle's program was installed in MACSYMA, the 1973 problem was solved as a test; the computer confirmed the results of the three-month calculation in two minutes. The experience is not unusual among users of computer-algebra systems.

In order to understand the need for automatic systems of algebraic manip

ulation it must be appreciated that many concepts in science are embodied in mathematical statements where there is little point to numerical evaluation. Consider the simple expression 3TT2/TT. As any student of algebra knows, the fraction can be reduced by canceling TT from both the numerator and the denominator to obtain the simplified form 3TT. The numerical value of 3TT may be of interest, but it may also be sufficient, and perhaps of greater utility, to leave the expression in the symbolic, nonnume rica I form. With a computer programmed to do only arithmetic, the expression 3TT2/TT must be evaluated; when the calculation is done with a precision of 10 significant figures, the value obtained is 9.424777958. The number, besides being a rather uninformative string of digits, is not the same as the number obtained from the numerical evaluation (to 10 significant figures) of 3TT. The latter number is 9.424777962; the discrepancy in the last two decimal places results from round-off errors introduced by the computer. The equivalence of 3TT2/TT and 3TT would probably not be recognized by a computer programmed in this way.

The example ill ustrates three advantages that algebraic programs have over

purely numerical ones. First, it is frequently more economical of computer time to simplify an expression algebraically before evaluating it numerically. Although in this example the saving in computer time is trivial, there are many complicated problems in which the economy that results from algebraic simplification is significant. Second, unlike the numerical approximations generated by a computer, algebraic answers are exact. Approximations necessarily introduce errors; if there are many successive numerical operations, the errors can accumulate and make nonsense of the final result. Only if a careful error analysis is undertaken can the final answer be stated with confidence, and such error analysis is one of the most complex problems faced in many fields.

The third and perhaps the most important advantage is that the goals of scientific investigation are often better served by a result in algebraic form. As Richard W. Hamming of Bell Laboratories has written, "the purpose of computing is insight, not numbers." Insight is sometimes obtained by evaluating a mathematical expression, but in many cases the relations of the quantities are made clearer by algebraic means.

In the study of a chemical process, for example, it is possible to express algebraically the relation between the stability of the process and the relative quantities of the substances present. From such a relation one can predict quite accurately whether a small change in one quantity will cause a violent reaction or a controlled one. Similarly, in the theory of stellar evolution one can examine algebraically how a number of variables determine whether a star will become a neutron star or a black hole, or how the variables interact to predict the existence of a new object.

There is a fourth advantage of the automatic systems we shall call computeralgebra systems, which is brought out by considering some pragmatic aspects of the accumulation of scientific knowledge. A scientific theory is often stated in a concise and quite general mathematical expression that suggests the

© 1981 SCIENTIFIC AMERICAN, INC

form the theory will take under certain assumptions. For example, a portion of the general theory of relativity, known as the Einstein field equations, can be written down in one line, given certain widely accepted notational conventions. To explore the physical implications of the theory is to explore the mathematical implications of the field equations, but with some exceptions the exploration requires algebraic operations that are intractable if they are done by hand. Even under assumptions that simplify the problem enormously, the algebra is so taxing that few investigators will risk their health, happiness and professional well-being attempting a solution.

The results of such work, even if they are admitted to the main body of scientific knowledge, stand as isolated intellectual outposts, untested by other workers and accepted primarily on the reputation of the investigator rather than on any promise of future critical examination. Such a regretta.ble state of affairs can prevail in any scientific field at what might be called the frontiers of intractability. For many fields computer algebra has now significantly expanded· the frontiers.

In celestial mechanics there is a distinguished tradition of long algebraic

calculations. In 1847 the French astronomer Charles Delaunay began to calculate the position of the moon as a function of time. In a sense the calculation is a straightforward application of Newton's theory of universal gravitation. In Newton's theory the orbit of a point mass around a spherical mass of uniform density is an ellipse, but the characteristics of the earth-moon system make the moon's orbit a curve considerably more complex than an ellipse. The plane of the moon's orbit around the earth is inclined at a small angle to the plane of the earth's orbit around the sun, and the angle varies under the perturbing influence of the sun's gravitational field. The sun also causes the lunar perigee (the point in the moon's orbit where it is closest to the earth) to precess slowly with respect to the stars.

Because of such complications the position of the moon has often been determined not from some function of time, as Delaunay set out to do, but by numerical extrapolation from a previous lunar position. Over short periods the errors accumulated in the extrapolation are small, and a correction can be made by again determining the position of the moon by observation. Calculating fast enough to keep up with the progress of the moon is fairly difficult even for the fastest computers, and the required observations are both time-consuming and costly. Delaunay's calculation avoided these difficulties, but it had one serious shortcoming: it demanded 20 years of his life to carry out and check.

The results of Delaunay's work were

a

b

c

d

e

g

2X3y4 + Oy2

zx(yx)2xzz

zxy2x2xzz

xx2xy2zzz

5y + 3x2 + 6y

- �x +;! Y 3 4

(c)

DELETE TERMS WITH ZERO COEFFICIENT )

EXPAND PARENTHESES )

ALPHABETIZE PRODUCTS )

WRITE IN EXPONENTIAL FORM )

ALPHABETIZE SUMS )

ADD COEFFICIENTS OF LIKE EXPRESSIONS )

WRITE OVER COMMON DENOMINATOR

)

+

+

+

+

_ 3XSy3z2 + 3xSy3z2 + X4y2z3 _ x4y2z3

4 2 3

3x2 + 5y + 6y

+

-ax + 9y 12

COMPUTER ALGORITHM for simplifying an algebraic expression by combining like

terms closely resembles the standard manual method of simplification. For the computer each

step of the procedure must be precisely specified. In the upper diagram the effect of each step

on a simple algebraic expression is illustrated. In the lower diagram the application of �he algo

rithm to the more complicated expression shown in color is illustrated by a flow chart. The col

ored letters that label the arrows in the flow chart correspond to the elementary operations in

the upper diagram. The simplified form of the expression generated by the algorithm is given

in color at the bottom. Simplification is one of the tasks that must be done by any set of comput

er programs intended to solve mathematical problems expressed in symbols rather than numer

ical quantities. Although the simplification algorithm usually yields a useful and manageable

algebraic expression, no single algorithm can reduce every expression to its simplest form.

137


(Cl) F(J): :FACTOR (NUSUM (K' J, K ,O,N));

(01) F(J) : '" FACTOR (NUSUM(KJ, K, 0, N)

(C2) FOR J: 1 THRU 2 DO DISPLAY (SUM(KtJ ,K,O,N) f IJ ) );

(02 )

> I

!< :: 0

!II (N + 1)

:.J (N + 1) (2 N + 1) > I

K = 0

(el) {X+7)'10;

(01) (X + 7)10

(e2) EXPAND (01) i (02 ) 2205 x8 + 41160 x7 + 504210 x6 + 4235364 x5

t 24706290 X4 + 98825160 x3 + 259416045 x2 + 403536070 X + 282475249

(el) X' 3+B*X' 2+A' 2*X '2-9 *A"'X '2+A' 2*B"'X-2 "'A"'S"'X-9"'Ai 3"'X+14"'1\ i 2*X-2 *A' 3*S+14*A '4"'0 i (01) x3 + B x2 + ,..2 x2 - 9 ,.. x2 + 1\2 B X - 2 A B X

- 9 ,..3 X + 14 1\2 X - 2 1\3 B + 14 ,..4 = 0

(e2) SOLVE(Dl,X);

102 )

(01 )

(e2)

T ime=

(02 )

-

IX = 7 A - B. X '" _ A2, X • 2 Al

11 25899906842624 y40 + 9007199254740992 y39 .. 43523068273885184

+ 96686654500110336 y37 + 71892942171668480 y36 _ 203545990580404224

- 3231739551940608 y3' + 2967153761027358720 y33 + 3933037175129505792

- 10392801849559220224 y31 - 26535490294760079360 y30 + 6097" 86 7 8 4 0 76 3 5599 36

+ 124971270109665427456 y28 _ 294061554052926275584 y27 _ 377892103185282105344

+ 1043007763435371888640 y25 + 1247934126027765186560 y24 _ 4589204662544561602560

- 534479284997751570432 y22 + 13715332431765689073664 y21 - 13302644475428239835136

- 11542363148164431609856 y19 + 30184905793343524306944 y18 _ 11553929011986625003520

- 20761961484788844621824 y16 + 23716893785532021784576 y15 _ 1470688776981600935936

- 11829726550983847370752 y13 + 7047410412446462752768 y12 + 59103 64 8 3 2 4 2 72 550133760

- 2234695017399712874496 y10 + 83656057" 103 2 70 3 54 4 4 2 2 4 y9 + 383" 8661311225029504

- 113171442907550345472 y7 + 29418059499453088640 y6 + 66680783706292864

- 1434405693611092512 y' 283861281845610304 y3 20855579198349088

+ 373175331659648 V 188618697410363

fACTOR (Dl) ;

130855 msec ,

( 4096 y10 + 8192 y9 _ 3008

221360 y' 1232 y3 + 144464

'" (4096 yl0 + 8192 y9 + 1600

135934 y' 18544 y3 + 11888

1< (8192 y13 ... 12288 y9 + 66560

90496 y' 356032 y3 + 113032

� (8192 V10 + 20480 y9 + 58368

414848 y' 4160 y3 + 171816

y8 _ 30848 y 7 + 21356

y2 _ 78488 V + 11993)

y8 - 20608 y 7 + 20032

y2 - 3416 Y + 41)

y8 _ 22528 y 7 _ 138240

y2 + 23420 Y - 8179)

y8 _ 161792 Y 7 + 198656

y2 48556 Y + 469)

y6 + 146496 y�

y6 + 87360 y5

yb + 572928 1'5

y6 + 199680 y;

y38

y35

y32

y29

y26

y23

y20

y17

yl'

y11

y8

y5

y2

COMPUTER-ALGEBRA SYSTEM called MACSYMA, developed at the Massachusetts Institute of Technology, is a general-purpose set of programs that can solve prohlems in algehra far beyond the abilities of most mathematicians. The system communicates with the user in a conversational style. After logging in to the system the user types the algehraic expression to be manipulated on the C line, or command line, in a computer language that resembles ALGOL or FORTRA:-;. The system responds on the D line with a rewritten version of the expression that is closer to the way it would appear on a printed page. The user then types a second command, and the computer carries out the indicated operation. In the first two examples the computer finds an expression for the sum of the first n terms of a series, where each term has the form of the expression next to the letter I. (Ik2 is a way of writing 12 + 22 + 32 + ... + 1/2.) In the next two examples the computer expands.a binomial expression and solves a nonnumerical cubic equation exactly. The upward-pointing arrows indicate exponentiation and ·the llsterisks indicate multiplication. The final example illustrates some of the computational power of the system. The maximum area of a hexagon inscribed in a circle with a diameter of 1 is one of the roots of the enormous 40th-degree pOlynomial shown. Manual methods of finding roots are unable to cope with a polynomial of this size; it is easier to factor the polynomial and find the roots of the factors. The computer-algebra system gives the most complete factorization whose terms have integral coefficients. A computer can solve the problem in about two minutes.

138

published in two volumes in 1867, and they remained unchallenged for more than 100 years. During World War II there was interest in expressing the coordinates of fixed stars as a function of time in order to help in determining the positions of German U-boats, but there was no further work on the orbit of the moon or of other bodies in the solar system. With the advent of artificial earth satellites, however, Delaunay's method for determining orbits became economically attractive.

In order to devise algebraic expressions for satellite orbits, Andre Deprit, J acq ues Henrard and Arnold Rom, who were then at the Boeing Scientific Research Laboratories in Seattle, began to investigate algorithms for doing the Delaunay calculation with a computer. In 1970 they made the calculation in about 20 hours of computer time. Remarkably, they found only three errors in Delaunay's work, all in terms of small value; moreover, two of the errors were mere consequences of the third one. Computer-algebra systems have since been developed that extend Delaunay's method by taking into account the effects of atmospheric drag and the nonspherical shape of the earth on a satellite in a low-altitude orbit. The algorithms are now sometimes employed in satellite tracking.

Considering the human life span, a reduction in the time needed for an operation from 20 years to 20 hours is a qualitative gain. Problems as complex as Delaunay's problem can become the focus of research for an investigator who has far less patience than Delaunay had. Moreover, the investigator can spend a much greater proportion of his time weighing the significance of his algebraic results. He can experiment with elaborate calc ulations without betting his career on the correctness and the utility of the answer. Perhaps the most telling indication that computer algebra has shifted the frontier of intractability in celestial mechanics is that Delaunay's huge calculation has been undertaken as a test of new computer-algebra systems.

How can a computer be programmed to carry out algebraic manipu

lations? Many of the algorithms employed in computer algebra began as the basic procedures devised by earlier generations of mathematicians that are now taught to algebra students in high school and college. For example, the methods for solving equations in which a single unknown quantity is raised to the first or the second power have been known since the time of Hammurabi, about 1750 B.C. Unlike the student who may develop an intuitive but vague sense of how to proceed through such a solution, however, the computer must follow a rigorous procedure that sp�cifies at every point what the computer is to do.

Consider how a student with a mod-


NUMBER CRUNCHING INSIGHT ALGEBRAIC ALGORITHM

1 + 2 = 3 1 ,275 + 51 = 1,326 1,326 + 52 = 1 ,378 1 ,378 + 53 = 1,431

1 ,431 + 54 = 1 ,485 1 ,485 + 55 = 1,540

1 ,540 + 56 = 1 ,596 1 ,596 + 57 = 1 ,653 1 ,653 + 58 = 1 ,711 1,71 1 + 59 = 1,770 1,770 + 60 = 1 ,830 1 ,830 + 61 = 1 ,891 1 ,891 + 62 = 1 ,953 1,953 + 63 = 2,01 6 2,01 6 + 64 = 2,080 2,080 + 65 = 2,145 2,1 45 + 66 = 2,21 1 2,211 + 67 = 2,278 2,278 + 68 = 2,346 2,346 + 69 = 2,41 5 2,415 + 70 = 2,485 2,485 + 71 = 2,556 2,556 + 72 = 2,628 2,628 + 73 = 2,701 2,701 + 74 = 2,775 2,775 + 75 = 2,850 2,850 + 76 = 2,926 2,926 + 77 = 3,003 3,003 + 78 = 3,081 3,081 + 79 = 3,1 60 3,160 + 80 = 3,240 3,240 + 81 = 3,321 3,321 + 82 = 3,403 3,403 + 83 = 3,486 3,486 + 84 = 3,570 3,570 + 85 = 3,655 3,655 + 86 = 3,741 3,741 + 87 = 3,828 3,828 + 88 = 3,916 3,916 + 89 = 4,005 4,005 + 90 = 4,095 4,095 + 91 = 4,186 4,1 86 + 92 = 4,278 4,278 + 93 = 4,371 4,371 + 94 = 4,465 4,465 + 95 = 4,560 4,560 + 96 = 4,656 4,656 + 97 = 4,753 4,753 + 98 = 4,851 4,851 + 99 = 4,950

4.950 +100 = 5,050

1 +2+3 48+49+50+51+52+53 98+99+1 00

3 + 3 = 6 6 + 4 = 10

10 + 5 = 15 15 + 6 = 21 21 + 7 = 28

LSW I WRITE GENERAL FORM OF EQUATION

I 1 +2+3+ ... + N = AN2 + 8N + C I 28 + 8 = 36 36 + 9 = 45 45 + 10 = 55 55 + 1 1 = 66 66 + 12 = 78 78 + 13 = 91 91 + 1 4 = 1 05

1 05 + 15 = 1 20 1 20 + 1 6 = 1 36 1 36 + 1 7 = 153

1 53 + 18 = 171 1 71 + 1 9 = 1 90 1 90 + 20 = 21 0 21 0 + 21 = 231 231 + 22 = 253 253 + 23 = 276

276 + 24 = 300 300 + 25 = 325 325 + 26 = 351 351 + 27 = 378 378 + 28 = 406

406 + 29 = 435 435 + 30 = 465 465 + 31 = 496

496 + 32 = 528 528 + 33 = 561 561 + 34 = 595 595 + 35 = 630 630 + 36 = 666 666 + 37 = 703 703 + 38 = 741

741 + 39 = 780 780 + 40 = 820 820 + 41 = 861 861 + 42 = 903 903 + 43 = 946 946 + 44 = 990 990 + 45 = 1,035

1,035 + 46 = 1,081 1,081 + 47 = 1,128 1 ,128 + 48 = 1,176 1 ,176 + 49 = 1 ,225

1 ,225 + 50 = 1,275

1 01

1 01

101

1 EVALUATE GENERAL EQUATION

FOR THREE VALUES OF N

IFN = 1, 1 = A(1 )2 +8(1) +C

IFN = 2, 1 + 2 = A(2)2 + 8(2) + C

IF N = 3, 1 + 2 + 3 = A(3)2 + 8(3) + C

1 SOLV E THREE EQUATIONS

IN THREE UNKNOWNS

H BHe,} A = V2 4A+28+C= 3 8 = V2 9A+38+C= 6 c=o

1 SUBSTITUTE IN GENERAL

EQUATION FOR A, 8 AND C

1+2+3+ ... + N =

'i2N2 + '/2N + 0 =

V2N(N + 1)

1 SOLVE EQUATION

FUR N = 100

1 +2+3+ ... + 1 00 =

'/2 X 1 00 x 101 =

5,050 1 01

50 x 101

~ 1 I 5,050 I

APPLICATION OF COMPUTER ALGEBRA to a numerical calculation can save computer-processing time by simplifying an algebraic expression before it is evaluated numerically. To find the sum of the first 100 integers a numerical calculation requires 99 separate additions. A simpler algorithm employed by the computer-algebra program solves the problem by taking advantage of a general mathematical result showing that the sum of the first II integers is a quadratic, or second-degree, polynomial function of II. Similarly, the algorithm would find the sum of the squares of the first II integers by constructing a third-degree polynomial function of II. Although the

algorithm is effective, it is not necessarily the fastest way to solve the problem or the one that leads to the clearest understanding of it. According to legend, the German mathematician Carl Friedrich Gauss noticed at the age of seven that the integers from 1 to 100 can be grouped in pairs so that all the pairs add to the same number, namely 101. The sum of the first 100 integers is therefore equal to 101 multiplied by SO, which is the number of pairs. Computer-algebra systems usually cannot recognize such patterns; on the other hand, Gauss's method cannot be applied to finding the sums of squares, cubes or higher powers of integers, as the computer-algebra algorithm can.

139


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erate facility for simple algebra might solve the equation 3x - 1 = 2(x + 5). First he would "clear parentheses" by multiplying both terms within the parentheses, x and 5, by 2, obtaining 3x - 1 = 2x + 10. He would then transfer the term 2x to the left side of the equation and the term - 1 to the right, changing signs as he goes and adding or subtracting terms as necessary. Thus in a single additional step he would obtain the answer x = 1 1.

In a computer-algebra program the steps to the solution of the problem have to be specified in somewhat tedious detail, and the shortcuts that might be recognized by a student during work on a particular problem must be spelled out. The first step in solving the equation in our example wo uld probably not d iffer in the program from the student's method: the program would clear parentheses and obtain the expression 3x - 1 = 2x + 10. In order to reach the last line, however, the program would have to transfer the terms of the equation one by one to a convenient side of the equal sign. One way to do this would be to program the computer to add an expression called the additive inverse to all the numerical constants appearing on the left side of the equal sign and to all terms that include variables appearing on the right side of the equal sign. Since - 1 is the only constant to appear on the left of the equal sign, the computer would add its additive inverse, + 1, to both sides of the equation, obtaining 3x - 1 + 1 = 2x + 10 + 1. The program would then rearrange the terms, if necessary, and add or subtract where possible, obtaining 3x = 2x + 1 1.

Next, the program would search the expressions on the right of the eq ual

sign for terms including variables and find the additive inverse of the first such term it encountered. In the example the computer would construct the additive inverse of 2x (namely -2x) and add it to both sides of the equation, thereby obtaining 3x - 2x = 2x + 1 1 - 2x. Once again the terms would be rearranged so that like terms could be combined: 3x - 2x = 2x - 2x + 1 1. Finally, the program would combine the terms having the same variable names and stop when a single variable appeared on the left side of the equation and a numerical constant appeared on the right: x = 11. Such an algorithm is neither the simplest nor the most efficient method one can devise for programming a computer to solve equations that have one unknown quantity, but it is not sufficiently complex either. There are many possibilities, such as the appearance of the expression 2x = 22 at some step in the solution of the problem, that the algorithm as we have described it is not equipped to deal with.

In developing an algorithm for computer algebra it is not necessary to fol-

low the procedure that is most efficient in manual calculation. Delaunay proved several theorems that he then employed to simplify intermediate calculations. Although the theorems could be incorporated into a program, the effort necessary to express them in algorithmic form encouraged Deprit and his colleagues to search for a method more compatible with mechanized execution. The one they invented requires transformations that would have exceeded the abilities even of Delaunay, but the algorithm is easy to program and can be exec uted quickly with a computer. The development of new algorithms is one of the most active areas of investigation in computer algebra; it is largely because of such work that computer-algebra systems have been improved significantly in the past few years.

In order to represent an algebraic expression in a computer program,

most systems store the minimum information needed to specify the expression uniquely. Current representational schemes employ one of two basic approaches or a combination of'the two. In one approach an expression is represented as an inverted treelike structure in which the leaves are the operands. For example, suppose one wished to represent the expression 2(x + 4) in a computer. The leaves of the tree would be the terms 2, x and 4, although they would appear at d ifferent levels. Both the x and the 4 would be connected by upward-moving branches to a plus sign. The symbol 2, however, would not be linked to the plus sign. Instead branches from the 2 and from the plus sign would

a

j+\ b

c

/y�;/\

In [Ian (x + i)] - sin h-1 [Ian (2x)1

SIMPLIFICATION of an expression presents a problem of choice for a corn puter-algebra system: given several equivalent forms of an expression, such as those in the upper part of the illustration, which is to be considered the simplest? Many systems now being designed leave the choice partly to the person working with the system. The complex expression in the lower part of the illustration is equal to zero for all values of x for which the tangents are positive real numbers. The expression does not vanish for all values of x, however. If a computer-algebra system were to substitute zero for the expression, the relation of the final answer to the original problem could become quite obscure. The expressions In, tan and sinh -1 are the standard notations respectively for the natural logarithm, the tangent and the inverse hyperbolic sine.

meet at the top, or root, of the tree, which would be labeled with a multiplication sign. The representation makes more efficient the search for subordinate expressions of predetermined form.

In the second scheme "slots" are assigned in some definite order to represent the information carried by an expression. To represent a polynomial in one variable, for instance, one slot is assigned to the name of the variable, the next slot to the degree of the polynomial (the largest power of the variable that appears in the polynomial) and the fol-

x 3 2 3 II -I 2x3 + 3x2 + Ilx - I

y y y

2

3 11 -I

4 /+,\ LEAVES /X\ 8 C �3 E

2 0

3y2 + 2y

2 -I

Ily + 2 -y -I

4(8 + C) + (A - 3E) 2x3 + (3y2 + 2y)x2 + (Ily + 2)x + (-y - 1) � 2x3 + 3X2y2 + 2x2y + llxy + 2x - Y - 1

REPRESENTING AN ALGEBRAIC FUNCTION in a computer may require several strategies. -The inverted tree diagram (a) is a simple and convenient means of representing the sequence of operations that apply to the variables and constants in an expression. A polynomial function of one variable can be represented by a sequence (b) that consists of the name of the variable, the degree of the polynomial (equal to the largest power of the variable that appears in the polynomial) and the numerical values of the coefficients of descending powers of the variable. To represent a polynomial function of two variables (c) a generalization of the polynomial representation for one variable can be employed. Here the horizontal group of slots represents a third-degree polynomial function of the variable x, whereas the vertical groups of slots represent polynomial functions of the variable y. The three polynomials in y are understood to be coefficients, or multipliers, of the terms of the polynomial in x that correspond to the three slots left blank. Hence the coefficient of the term x2 is 3y2 + 2y, the coefficient of the term x is 1 1y + 2 and the coefficient of the constant term of the pOlynomial in x is - y - 1.

143


QUESTAR: 12 on the QUESTAR MOUNT

The Questar 12, latest addition to the Questar family of fine optical instruments, now has its own Questar-designed mount. A German equatorial type, it is notable for its 3600 continuous tracking in A.A. with precision tracking to better than 4 arc seconds. The Questar Mount is designed with over-sized components so that it can accommodate any Questar up to 18 inches. The standard mount shown is straightforward in design but can be modified so as to be compatible with more sophisticated tracking devices or other special equipment.

The Questar 12 is a superb instrument for the serious astronomer, for the university astronomy department or the engineer seeking sophisticated tracking and surveillance equipment for which Questar Corporation has a noted reputation.

Questar Corporation Box C, Dept. 20, New Hope, Pa. 18938 (215) 862-5277

Observing domes suitable for housing the Questar 12 or larger telescopes are also available at Questar.

C Questar Corporation 1981

144

lowing slots to the coefficients of descending powers of the variable. The set of information slots can be made a part of a tree-structure representation when more complicated expressions must be stored.

Although every representation in a I\. computer must be unambiguous, any given algebraic expression can be represented in a variety of equivalent ways. One might want to construct an algorithm that would always represent an expression in the simplest form possible; in such an algorithm, for instance, x + x + x might always be represented as 3x. People disagree, however, over what constitutes the simplest form of an expression. Stylistic preferences differ in much the same way as political ideologies do: programs designed by conservatives make no transformations on an expression unless they are specifically instructed by the user to do so, whereas programs designed by radicals always change the user's expression to their own preferred form. Many people like to retain some control over the simplification of an expression because the utility of a particular form may depend on the circumstances in which it is encountered. In any event, even if people could agree on what is the simplest form of every expression, it is known that the construction of a general algorithm for simplification is impossible.

Computer-algebra systems can simplify expressions that are exceedingly large and complex. Such expressions may incorporate not only the elementary polynomial, trigonometric and logarithmic functions but also the more complex functions that can arise in scientific work. The user of a computer-algebra system can also define his own functions, specify their properties and

COMPUTE REMAINDER Rl WHENy + x

PRINT

GCO(x;y) = x

GCO( 102.131 ;251.591)

2 102,131 ) 251,591

204,262

Rl = I 47,329

Rl = O?

NO

FLOW CHART diagrams the operation of the Euclidean algorithm, a rigorous procedure for determining the greatest common divisor (GCD) of two integers. (The greatest


supply the rules of simplification appropriate to them; indeed, the computer can sometimes be employed to find such rules. In most cases little is known in advance about how an expression will simplify. The most practical method is to retrieve from the computer's memory all the kinds of algebraic expression the program can already simplify, supply whatever additional algebraic identities the programmer can think of and allow the computer to proceed on its own. The method works surprisingly well: it can often simplify as effectively as an expert mathematician.

Because of the need in algebraic calculations for exact rather than approximate answers, a computer-algebra system must be able to manipulate huge numbers with unlimited precision. In most computers numbers are ordinarily represented with a fixed number of significant digits, although the precision of the arithmetic operations can sometimes be doubled or even increased severalfold. For the purposes of computer algebra, however, no limits on precision can be set in advance.

Consider one algorithm that can be employed to simplify a rational polynomial expression (one in which one polynomial is divided by another). The algorithm is a generalization and modification of a method called the Euclidean algorithm for determining the greatest common divisor (OCD) of two numbers, which has been known for 2,200 years. The OCD of two numbers, such as 6 and 8, is found by a fixed sequence of repeated divisions; once the OCD is found, a fraction can be reduced to its lowest terms by dividing both the numerator and the denominator by the OCD. For the fraction 6/8 the OCD is 2 and the reduced form is 3/4.

When the Euclidean algorithm is ap-

COMPUTE REMAINDER R2 WHEN x + R1

PRINT

GCD(x;y) = R1

� 47,329

plied to a polynomial, however, the quotients and remainders in the repeated divisions can rapidly develop terms in which the fractional coefficients are ratios of enormous numbers. To avoid computing with enormous numbers, certain techniques have been devised for reducing the size of the numbers carried along in the computation. For example, a constant numerical factor can be eliminated from a polynomial in the Euclidean algorithm. Rounding any of the coefficients before the calculation is complete, however, would make nonsense of the final answer.

One of the most important algorithms developed in the past 15 years for

computer-algebra systems is an algorithm that can solve problems in indefinite integration. Integration is a basic method of calculus, first invented independently by Newton and Leibniz, whereby an unlimited number of arbitrarily small quantities can be combined to yield some specific quantity . In what is called indefinite integration, the method is generalized : the outcome of the integration is expressed as a function of at least one variable instead of as some definite quantity. Such problems are encountered often in the physical and biological sciences, but their difficulty has bedeviled mathematicians for hundreds of years. It was once thought that no general algorithm for the solution of such problems could be constructed; the standard approach to their solution calls both for guesswork and for consulting published tables of integrals.

The algorithm that was successfully constructed for indefinite integration is similar to an algorithm subsequently developed to determine an algebraic expression for a finite sum of expressions that all have the same form [see illustra-

COMPUTE REMAINDER R3 WHEN R1 + R2

tion on page 139]. If an integral can be solved at all in closed form (rather than as the sum of an infinite series of terms), it is possible to predict the general algebraic form of the solution and then work backward by differentiation (the reverse of integration) to determine the exact formula. The algorithm has been incorporated into several computer-algebra systems. A numerical computer study of e ight widely employed tables of indefinite integrals discovered that about 10 percent of the formulas are in error; one of the tables was found to have an error rate of 25 percent.

The funds spent on the development of computer-algebra systems probably do not exceed $3 million per year in the U.S. Nevertheless, the systems already exhibit great sophistication and diversity of design. There are about 60 such systems today, and they can be classified into three main groups that reflect their historical development. The systems in the first group are the descendants of the earliest attempts to write programs in computer algebra; they were designed to solve specific problems in fields such as mathematical physics and theoretical chemistry. Because a special-purpose program can be finely tuned for the kind of inp ut expected, it can operate at great speed. Currently such programs exist for solving problems in quantum electrodynamics (ASHMEDAI), lunar theory and general relativity (CAMAL), high-energy physics (SCHOONSCHIP), indicial tensor manipulation (SHEEP) and celestial mechanics (TRIGMAN) and for solving equations limited to polynomials and rational functions (ALTRAN). There is even a special-purpose program called ALDESsAc/2 that can assist in developing new computer-algebra programs.

The systems in the second major

COMPUTE REMAINDER R4 WHEN R2 + R3

�L- _

R_4

_=

_0_

? __ �

I PRINT L.J:::\ L

__ G_

C_D

_�

_; y

_)

_=

_R

_3

_ --J1 �� PRINT

GCD(x;y) = R2

3

7,473 7,473

R4= G

R4 = O?

YES

GCD(102, 131;

2 51.591) = 2 ,491

STOP

common divisor is the largest integer that will divide both given integers without leaving a remainder.) The algorithm was discovered at least 2,200 years ago; except for the four elementary operations of arithmetic it is the oldest algorithm known. Colored arrows in the

flow chart indicate the transitional states of the algorithm for the numerical example in the lower part of the diagram. The Euclidean algorithm is not limited to integers; a slightly modified procedure can determine the greatest common divisor of two polynomials.

145


group are the general-purpose ones, which provide the investigator with as many mathematical capabilities as possible. Such systems can carry out the four basic arithmetic operations, perform definite and indefinite integration and solve equations, including ordinary differential equations that express algebraically how the change in one variable

depends on changes in other variables. The programs can also solve systems of linear or nonlinear algebraic equations, differentiate, simplify, factor, compute with finite or infinite sums of expressions called Taylor series and perform all the functions of the special-purpose systems, although not with as much speed and power. The four best-known

P = XS + 15x' + 85x3 + 225x2 + 274x + 120

o = 6x' + 77x3 + 304x2 + 363x + 90

GCO(P;O)

P+O

6x' + 77x3 + 304x2 + 363x + 90) XS + 15x' + 85x3 +

R1 = O?

NO

0+ R1

XS + 15x' + 2.825x3 +

Rl= 235x3 +

10 152x + 45 180 11,045

235x3 + 1,970x2 + 4,605x + 3,150 )8x' + 77x3 + 36

225x2 +

6 130x2 +

1.970x2 +

304x2 +

6x4 + 77x3 + 754, 192 x2 +

R2 = O?

NO

R1 + R2 R2 = _ 82.656x2 +

274x +

5,259x + 36

4.605x +

36

363x +

1 ,333,515x + 2,209

531.648x + 2,209

_ 148 986 005x + 290 660 220 854,001,792 ) 235x3 + 1 970x2 + 4,605x

36

120

1,170

3,150

90

790,650

591,840

+ 3,150

235x3 + 1.970x2 762.992,310x + 540,793,800 36

+ 5,930.568 R3 = O?

NO

R2 + R3

_ 121,495 (x + 5) 164,738

R4 = O?

YES

GCO(P;O) = x + 5

R3 = 121,495

- 164,738 (x + 5)

13.616.584,128x + 19,499.707.584 268,382,455 ) _ 82,656x2 + 531.648x + 591,840

2,209

82,656x2 + 531.648x + 591,840 2,209

R4= G EUCLIDEAN ALGORITHM applied to polynomials determines their greatest common divisor by repeated divisions, in the same way as it determines the greatest common divisor of two integers. The algorithm can give rise to huge intermediate numerical results that cannot be rounded off. For this reason the arithmetic operations in a computer-algebra system must have unlimited precision. Ordinarily a number stored in a computer is allotted a fixed amount of space in the computer memory, but in a computer-algebra system no fixed allocation can be made. Mathematicians who study the properties of large numbers have been attracted to this feature of computer-algebra systems. Colored arrows and the statements printed in color correspond to the steps of the algorithm diagrammed in the illustration on pages 144 and 145.

146

general-purpose systems are MACSYMA; REDUCE, a system developed at Stanford University, the University of Utah and the Rand Corporation; SCRATCH PAD, developed by the International Business Machines Corporation, and SMP, developed at the California Institute of Technology. They represent the most advanced achievements in computer algebra so far.

Computer-algebra systems designed to operate with microcomputers have now begun to appear. They are slower and less comprehensive than the general systems designed for large computers, but they can still perform far more complex calculations more accurately than many mathematicians can. The most sophisticated and widely available such system is called mUMATH; it was developed at the Soft Warehouse in Honolulu. MUMATH provides some of the capabilities of the general systems, although the size, memory and speed of the microcomputer do not allow mUMATH to attack very complex problems. Nevertheless, such systems (or their more powerful descendants) may find their way into personal microcomputers and perhaps even notebook-size calculators before the end of the decade.

Computer-algebra systems tend increasingly to be written in simple,

high-level programming languages that resemble ordinary mathematical expressions. The systems are also generally designed to interact with the programmer in a conversational manner. The beginner can often learn in a few minutes to solve many kinds of nontrivial problem beyond his power to solve by hand. With most general systems the user types in a command in a language similar to the computer languages ALGOL or FORTRAN. If the command includes fractions, exponents or other symbols that do not normally appear on the same line in print, the computer replies with a rewritten version of the expression that d isplays it in more natural form. The user can then specify certain operations to be carried out on the expression, such as adding it to itself, raising it to a power, differentiating it or integrating it. The computer executes the operation, simplifies the resulting expression and prints it or displays it on a video terminal. If the operation depends on, say, whether the positive or the negative square root of an expression is to be extracted, the computer queries the user before the calculation is finished.

Computer algebra has been applied in a variety of disciplines, including acoustics, algebraic geometry, economics, fluid mechanics, structural mechanics and number theory, and in the design of propellers, ship hulls, helicopter blades, electron microscopes and large-scale integrated circuits. We shall describe three additional applications in which com-


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A major breakthrough in seismology is increasing our success ratio in finding new energy reserves.

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Everything about David Plastowrepresents his companys philosophy. Which is why he wears a Rolex.

David Plastow is the custodian of a long and famous engineering tradition.

He is the Chief Executive of Rolls-Royce Motors , and indeed, his manner and personal appearance exact-1y reflect the ethos of that company. That of the skilled engmeer.

Plastow takes a personal interest in any modification, however small .

"All our developments at Rolls-Royce are always evolutionary rather than revolutionary," he says .

"We are a highly personal business , and both our craftsmen and our customers have clearly defined ideas about what a RollsRoyce should be. But while we don't tamper with those fundamental ideas , we are , of course , constantly searching for improvement. For instance, years ago, the gear selection on a Rolls-Royce car became completely electronic. But, a driver likes to 'feel'

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David Plastow recognizes the similar philosophy behind the watch he wears .

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puter algebra is making contributions. For investigations in plasma physics

and the development of fusion-energy sources some calculations require numerical and algebraic computer techniques to be combined. The algebraic formulas that describe the properties of a plasma in a magnetic field can become quite complex, and they tend to impede physical understanding. What has been lacking is a means for obtaining approximate rather than exact analytic expressions for such phenome'na. To solve the problem the Plasma Theory Group at

M. I.T. has devised a computer-algebra technique capable of partitioning the terms in the computation according to the physical processes thal generate them. By numerical methods one can then eliminate from the series of terms describing each process those that are relatively small. The result is an algebraic expression that describes the dominant properties and remains sufficiently simple for the physical significance of each term to be perceived.

In order to derive testable predictions from theories of the interactions of ele-

Ja fonction R ne contient plus alleun terme periQdique ; e1le se trouve dont: reduite it son temle non periodique selll , tel'me qni, en tenant ('ompte des parties fOllrnics par Ies operations I �91 �6o l 349 et 4 1 5, a pour valeul"

- (t;

MASSIVE ALGEBRAIC CALCULATION completed by hand in 1867 was recomputed for the first time in 1970 with a computer-algebra system. The original calculation was undertak. en by the French astronomer Charles Delaunay and was published in two volumes; a page of the second volume is reproduced here. Delaunay's calculation, which took 10 years to fin. ish and another 10 years to check, gave the position of the moon as a function of time to a preci.

sion never before attained. Three errors were detected when the calculation was checked by

Andre Deprit, Jacques Henrard and Arnold Rom of the Boeing Scientific Research Laborato

ries in Seattle. The major error, which gave rise to the other two errors, is outlined in color; the

expression should be 3 3 / 1 6 y2e'2. The checking required about 20 hours of running time on a

computer. Although computer algebra was responsible for exposing Delaunay's errors, the tao bles have since been turned: the remarkable precision of the calculation has been employed

to test a few new computer-algebra systems for accuracy before they were put into service.

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mentary particles it has been necessary to evaluate expressions that can include thousands of integrals, many of which have unusual and deceptive mathematical properties. The numerical evaluation is open to two of the criticisms of numerical methods we have already mentioned: it is not as useful as an algebraic result for showing how contributions from various physical phenomena affect the interaction, and it introduces errors of approximation_ As experimental techniques are refined the uncertainties introduced by approximation can become so large that the predictions of the theories cannot be distinguished. It is likely the application of computer-algebra methods will be of great help in generating theoretical predictions of the required precision.

In the study of theoretical alternatives to general relativity a test called Bir k hoff's theorem has recently become popular . The mathematician George David Birkhoff of Harvard University showed in 1923 that the general theory of relativity excludes the propagation through space of gravitational p ulses that might conceivably be generated by radial pulsations of matter in a star. Be cause no such pulses have been detected . and because they are impossible according to both Einstein's and Newton's the ories of gravity, the exclusion of the pul ses has come to be expected of any potential gravitational theory. The calculation req uired to determine whether or not a theory of gravity satisfies Birkhoff's theorem is lengthy, but it has now become straightforward with computeralgebra systems. A gravitational theory formulated by C. N. Yang of the State University of New York at Stony Brook was recently shown by the use of a computer-algebra system to violate Birkhoff's theorem. The calculation corrected a hand calculation that had been believed and cited for many years.

I t is important to recognize that computer-algebra systems need not be

employed solely for large calculations. Relatively simple problems often arise for which algebraic manipulation can contribute to understanding or make numerical calculation more efficient. One can sometimes reduce numerical computer-processing time by a factor of 100 or more by the j udicious application of algebraic processing. Moreover, computer-algebra programs can substitute for an entire library of mathematical reference works. "Knowledge-based" programs that include integration methods, methods for solving differential equations and the like may make such reference works as obsolete as a table of logarithms. As inexpensive computers are improved, computer algebra will become available for teaching, study, research and perhaps unthought-of applications to all interested people in their own offices and homes.


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Computer Algebra

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Transcript of Computer Algebra